158 Dr. J. W. Nicholson on the Bending of 



the zero point in the continuation of that region is still 

 determined from 



zv = <j> n —<j) n r—m0 } (67) 



while it exists, and we may write, with this value 



ry = X"u(l + e 2l Xn)e l * v , 



where on reduction, 



u= -e* l7r {JcR n R nr I 2 Trar 2 sin 6)K . . (68) 



In this formula m = z has been inserted, for they now 

 begin to differ by a quantity of lower order than z, and u is 

 not oscillatory. The zero point, except in oscillating 

 functions, is now sufficiently given by 0=1. But Xn can 

 no longer be written as zero, for when 0=1, it reaches the 

 value ^7r. At the zero point it has a certain finite value ^ 

 not quite of order unity in this case. Again, under the same 

 circumstances, 



V = V + ±V " {X-XqY, 



neglecting higher terms near the zero point «r . 



The most significant part of the magnetic force becomes 

 therefore 



y = u (l + e 2l x°)ze l * Vo f 



*Z 



b zv o"Z 2 



= u (l + 4*x*){2irzlv n )te uv °+t 1 *, . . (69) 



but its further examination is not important. It is a solution 

 of the same type as in the region of brightness, but no longer 

 represents the oscillator in the presence of a plane reflector. 



But on passing further into the transitional region, with 

 6 still not small, 7)<f) n fdn and "d<j) n r/?)n become of order m~a, 

 and can no longer balance 6. The zero point therefore 

 ceases to exist, and we enter upon the region more truly 

 defined as that of transition in which we may expect, from 

 ordinary considerations, to find bands of alternate maximum 

 and minimum intensity, which, on the other side, merge 

 into a continuous shadow. These bands will now be shown 

 to occur, but before ~d<f> n fon is of order m~£. 



In order that a set of harmonic terms in the series should 

 become of supreme importance, it is not necessary that the 

 derivate of an exponent should be zero, but only that it be 

 small. Hitherto it has not been possible for this to occur 

 without the zero value being attained. But in the present 

 case, we have just passed from a region in which the derivate 

 can be zero to one in which it cannot. Although v' cannot 

 now become zero, it must attain a certain minimum value e 



